Non-Unitary Quantum Machine Learning: Fisher Efficiency Transitions from Distributed Quantum Expressivity

Abstract

Quantum machine learning has faced growing scrutiny over its practical advantages compared to classical approaches, particularly following dequantization results and large scale benchmarking studies that have challenged earlier optimistic claims. This work presents a systematic empirical evaluation of non unitary quantum machine learning implemented via the Linear Combination of Unitaries framework within hybrid quantum classical neural networks. Across more than 570 experiments spanning four domains digit classification MNIST, agricultural disease detection PlantVillage, molecular property regression QM9, and medical histopathology PathMNIST non unitary quantum layers are benchmarked against structurally identical unitary baselines. Consistent performance improvements are observed across all domains, with gains ranging from +0.2 percentage to +5.8 percentage depending on dataset complexity and qubit count. A particularly notable finding is a Fisher efficiency transition in medical imaging tasks, where parameter efficiency shifts from negative to positive as qubit count increases from 10 to 12, indicating a threshold dependent efficiency regime. Additionally, non unitary IQP circuit variants reach or exceed classical baselines at 10 qubits on CIFAR 10, demonstrating that circuits with established complexity theoretic hardness guarantees remain compatible with competitive learning performance under the LCU framework. These results offer a large scale, evidence based characterisation of the conditions under which non unitary QML yields measurable empirical benefits in near term settings.

Figures (4)

• Figure 1: Non-unitary quantum layer (LCU) performance across qubit scales for three CNN architectures on MNIST. All architectures show consistent performance improvements relative to matched baselines that persist or increase with qubit count. Error bars represent standard deviation across 10 runs.
• Figure 2: Fisher efficiency transition in PathMNIST at 12 qubits. Non-unitary quantum layers (LCU) achieve comparable performance to unitary baselines (NoLCU) while demonstrating superior parameter efficiency, crossing from negative to positive Fisher efficiency. This transition indicates that non-unitary transformations enable equivalent or better performance with effectively fewer quantum parameters.
• Figure 3: Non-unitary IQP circuit scaling on CIFAR-10. Both IQP Layer (single IQP for variational layer) and IQP Embedding (double IQP for encoding and variational), incorporating non-unitary transformations via LCU, show favorable scaling, crossing the classical baseline at 10 qubits. The threshold behavior suggests minimum quantum resource requirements for non-unitary IQP advantages, while the variance reduction with qubit count indicates improved circuit stability.
• Figure 4: Favorable scaling in PlantVillage agricultural disease classification. Non-unitary quantum layer (LCU) advantages increase from +4.91% at 10 qubits to +5.78% at 16 qubits, while unitary baseline (NoLCU) improvements saturate. Non-unitary models scale approximately 40% better than unitary quantum approaches, suggesting that performance remains stable within the evaluated qubit range.